Rings of continuous functions vanishing at infinity on a frame

What is it about?

Let $ \mathcal{R}_{\infty}(L)= \Bigl\{\, \varphi\in \mathcal{R}L : \mbox{ ${\uparrow}\varphi ( \dfrac{-1}{n}, \dfrac{1}{n})$ is compact for each $n \in\mathbb{N}$} \,\Bigr\}$ that was introduced by Dube [2010] in `` On the ideal of functions with compact support in point-free function rings, Acta Math. Hungar., 129(3) (2010), 205-226" and next by Guti\'errez Garc\'ia et al. [2014] in ``Notes on point-free real functions and sublocales, Categorical methods in algebra and topology, 167-200, Textos Mat./Math. Texts, 46, Univ. Coimbra, Coimbra, 2014. This is the frame analogue of the subset of C(X) consisting of functions that vanish at infinity (see ``Rings of continuous functions, Springer-Verlag, New York, 1976.) and also, maximal ideals $ \mathcal{R}_{\infty}(L)$ are studied in ``On maximal ideals of $\mathcal{R}_{\infty}L$, Journal of Algebraic Systems, 6(1) (2018), 43-57.

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Why is it important?

In functional analysis, an important role is played by the ideal Co(X,C ) of all functions that vanish at infinity. Co(X) is the intersection of all the free maximal ideals in C*(X).

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